
In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός ''homos'' "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups. For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces. The original motivation for defining homology groups is the observation that shapes are distinguished by their ''holes''. But because a hole is "not there", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. It approaches the problem through the idea of ''cycles''  closed loops or lowdimensional manifolds  that can be drawn on a given n dimensional manifold but not transformed smoothly into each other, for example because they pass through different holes. As it turns out, subtle kinds of holes exist that homology cannot "see" — in which case homotopy groups may be what is needed. == Informal examples == Informally, the homology of a topological space ''X'' is a set of topological invariants of ''X'' represented by its ''homology groups'' :$H\_0(X),\; H\_1(X),\; H\_2(X),\; \backslash ldots$ where the $k^$ homology group $H\_k(X)$ describes the ''k''dimensional holes in ''X''. A 0dimensional hole is simply a gap between two components, consequently $H\_0(X)$ describes the pathconnected components of ''X''. A onedimensional sphere $S^1$ is a circle. It has a single connected component and a onedimensional hole, but no higherdimensional holes. The corresponding homology groups are given as :$H\_k(S^1)\; =\; \backslash begin\; \backslash mathbb\; Z\; \&\; k=0,\; 1\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$ where $\backslash mathbb\; Z$ is the group of integers and $\backslash $ is the trivial group. The group $H\_1(S^1)\; =\; \backslash mathbb\; Z$ represents a finitelygenerated abelian group, with a single generator representing the onedimensional hole contained in a circle. A twodimensional sphere $S^2$ has a single connected component, no onedimensional holes, a twodimensional hole, and no higherdimensional holes. The corresponding homology groups are〔 :$H\_k(S^2)\; =\; \backslash begin\; \backslash mathbb\; Z\; \&\; k=0,\; 2\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$ In general for an ''n''dimensional sphere ''S^{n}'', the homology groups are :$H\_k(S^n)\; =\; \backslash begin\; \backslash mathbb\; Z\; \&\; k=0,\; n\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$ A onedimensional ball ''B''^{1} is a solid disc. It has a single pathconnected component, but in contrast to the circle, has no onedimensional or higherdimensional holes. The corresponding homology groups are all trivial except for $H\_0(B^1)\; =\; \backslash mathbb\; Z$. In general, for an ''n''dimensional ball ''B^{n}'',〔 :$H\_k(B^n)\; =\; \backslash begin\; \backslash mathbb\; Z\; \&\; k=0\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$ The torus is defined as a Cartesian product of two circles $T\; =\; S^1\; \backslash times\; S^1$. The torus has a single pathconnected component, two independent onedimensional holes (indicated by circles in red and blue) and one twodimensional hole as the interior of the torus. The corresponding homology groups are :$H\_k(T)\; =\; \backslash begin\; \backslash mathbb\; Z\; \&\; k=0,\; 2\; \backslash \backslash \; \backslash mathbb\; Z\backslash times\; \backslash mathbb\; Z\; \&\; k=1\; \backslash \backslash \; \backslash \; \&\; \backslash text\; \backslash end$ The two independent 1D holes form independent generators in a finitelygenerated abelian group, expressed as the Cartesian product group $\backslash mathbb\; Z\backslash times\; \backslash mathbb\; Z$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Homology (mathematics)」の詳細全文を読む スポンサード リンク
